#386790

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,9,15,11,16,10,14)(7,17,19,29,21,28,20,12)(8,22,24,32,26,31,25,13)(18,30,34,36,23,27,35,33);(2,4)(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,30)(14,28)(15,17)(16,29)(18,26)(22,36)(24,35)(25,34)(27,32)(31,33);(1,2,3,4)(5,8,9,24,11,26,10,25)(6,13,14,31,16,32,15,22)(7,18,19,34,21,23,20,35)(12,27,28,36,29,30,17,33);(7,19,21,20)(12,28,29,17)(18,34,23,35)(27,36,30,33)

Polycyclic presentation:

2^9|a^2=h,c^2=fg,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293986);